We consider the problem of forwarding packets arriving online with their destinations in a line network. In each time step, each router can forward one packet along the edge to its right, and the packet arrives at the next router one time step later. Packets are forwarded until they reach their destination. The flow time of a packet is the elapsed time between its release and its arrival at its destination. The goal is to minimize the maximum flow time. This problem was introduced by Antoniadis et al. in 2014, with a focus on line networks. They proposed several natural algorithms. For one, they proved that it is not O(1)-competitive; for others, they claimed analogous lower bounds, seemingly leaving no natural candidate for an O(1)-competitive algorithm. In this paper, we study a natural algorithm not considered in that work. Our algorithm, simply called Greedy, selects packets according to their projected flow time under the assumption that they are not delayed any further. We focus on the special case in which each packet needs to be forwarded by one or two routers; this case captures core difficulties. We show that Greedy achieves a competitive ratio of exactly 2-2^1-k, where k is the number of active routers in the network. We also give the first nontrivial general lower bound, which applies even to randomized algorithms: using the same type of instances as in our lower bound for Greedy, we show that no algorithm can be (4/3-ε)-competitive for any ε>0.

We consider the problem of minimizing the weighted makespan on a single machine with restarts. Restarts are similar to preemptions but weaker: a job can be interrupted, but then it has to be run again from the start instead of resuming at the point of interruption later. The objective is to minimize the weighted makespan, defined as the maximum weighted completion time of jobs. We establish a lower bound of 1.46557 on the competitive ratio achievable by deterministic online algorithms. For the case where all jobs have identical processing times, we design and analyze a deterministic online algorithm that improves the competitive ratio to better than 1.3098. Finally, we prove a lower bound of 1.2344 for this case.

We consider the online buffer minimization in multiprocessor systems with conflicts problem (in short, the buffer minimization problem) in the recently introduced flow model. In an online fashion, workloads arrive on some of the n processors and are stored in an input buffer. Processors can run and reduce these workloads, but conflicts between pairs of processors restrict simultaneous task execution. Conflicts are represented by a graph, where vertices correspond to processors and edges indicate conflicting pairs. An online algorithm must decide which processors are run at a time; so provide a valid schedule respecting the conflict constraints. The objective is to minimize the maximal workload observed across all processors during the schedule. Unlike the original model, where workloads arrive as discrete blocks at specific time points, the flow model assumes workloads arrive continuously over intervals or not at all. We present tight bounds for all graphs with four vertices (except the path, which has been solved previously) and for the families of general complete graphs and complete bipartite graphs. We also recover almost tight bounds for complete k-partite graphs. For the original model, we narrow the gap for the graph consisting of a triangle and an additional edge to a fourth vertex.

In this column, we will discuss some papers in online algorithms that appeared in 2025. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

We consider the problem of minimizing the total completion time on a single online machine using restarts. Although restarts can potentially be very beneficial in the context of online scheduling, there has been relatively little research on this topic up until now. We present a very simple online algorithm which is better than 1.4568-competitive. The basic rule of the algorithm is to run jobs in order of increasing size. For possible restarts, the algorithm only considers whether the completion time of an incoming job is more than a factor of 1.4568 higher if we first let the running job finish compared to the case where we start the new job immediately. If this is the case, we interrupt the running job and start running the new job. All other existing or past jobs are ignored for this decision. The analysis of the algorithm has become significantly easier and shorter than for the previous best result which was 3/2. We hope that this result can lead to further research in this interesting topic.

We break the barrier of 3/2 for the problem of online load balancing with known makespan, also known as bin stretching. In this problem, m identical machines and the optimal makespan are given. The load of a machine is the total size of all the jobs assigned to it and the makespan is the maximum load of all the machines. Jobs arrive online and the goal is to assign each job to a machine while staying within a small factor (the competitive ratio) of the optimal makespan. We present an algorithm that maintains a competitive ratio of 139/93 < 1.495 for sufficiently large values of m, improving the previous bound of 3/2. The value 3/2 represents a natural bound for this problem: as long as the online bins are of size at least 3/2 of the offline bin, all items that fit at least two times in an offline bin have two nice properties. They fit three times in an online bin and a single such item can be packed together with an item of any size in an online bin. These properties are now both lost, which means that putting even one job on a wrong machine can leave some job unassigned at the end. It also makes it harder to determine good thresholds for the item types. This was one of the main technical issues in getting below 3/2. The analysis consists of an intricate mixture of size and weight arguments.

For this issue, I am pleased to present a tutorial by Will Ma on using randomized rounding to design online policies. Randomized rounding has of course been used extensively in the design of approximation algorithms and also to approximately solve sequential stochastic optimization problems. Will describes several examples that show how randomized rounding can be used in this context. As always, I would like to invite more contributions to this column, be it surveys, conference reports, or technical articles related to online algorithms and competitive analysis. If you are considering becoming a guest writer, don't hesitate to mail me at rob.vanstee@uni-siegen.de.

In this column, we will discuss some papers in online algorithms that appeared in 2024. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

In this column, we will discuss some papers in online algorithms that appeared in 2022. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

In this column, we will discuss some papers in online algorithms that appeared in 2023. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

In the discrete bamboo garden trimming problem we are given n bamboo that grow at rates v 1 , . . . , v n per day. Each day a robotic gardener cuts down one bamboo to height 0. The goal is to find a schedule that minimizes the height of the tallest bamboo that ever exists. We present a 10 / 7-approximation algorithm that is based on a reduction to the pinwheel problem. This is consistent with the approach of earlier algorithms, but some new techniques are used that lead to a better approximation ratio. We also consider the continuous version of the problem where the gardener travels in a metric space between plants and cuts down a plant each time he reaches one. We show that on the star graph the previously proposed algorithm Reduce-Fastest is a 6-approximation and the known Deadline-Driven Strategy is a (3 + 2 √ 2)-approximation. The Deadline-Driven Strategy is also a (9 + 2 √ 5)-approximation on star graphs with multiple plants on each branch.

For this issue, I am pleased to present an extensive survey by Debasis Dwibedy and Rakesh Mohanty on online makespan scheduling. This is a topic which continues to inspire new research and it is great that these authors have provided an updated survey.

In this column, we will discuss some papers in online algorithms that appeared in 2021. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

In the cake cutting problem, a cake needs to be divided among players with possibly different valuations for different pieces of the cake. We consider a variation where the cake represents chores that need to be allocated. We assume the players to be work-averse, i.e., intending to get as little of the cake as possible. We further assume that the chores are indivisible and that they are ordered along a line. We seek a fair allocation of the chores. We study the loss in social welfare due to fairness using the price of fairness concept. Previous work has considered fair division of cakes and chores with non-connected and connected pieces as well as fair division of indivisible contiguous blocks of indivisible items (with positive utility) on a line. We complete the picture by providing price of fairness results for the allocation of contiguous blocks of indivisible chores.

In the buffer minimization in multiprocessor systems with conflicts or simply buffer minimization problem, a multi-processor system is modelled as an undirected graph. A conflict occurs if two processors are connected by an edge. Conflicting processors can not run at the same time. At any time, load may arrive on one or more processors. Incoming workload is stored in an input buffer and a machine that is running reduces its workload at a constant rate. The goal is to find a schedule that minimizes the maximum workload over all machines. We consider the special case where the graph is a path and give bounds on the competitive ratio for small graph sizes, including a tight bound of 9/4 for a path with 4 nodes. We give a general lower bound of 12/5. We also consider online algorithms that have resource augmentation on their speed, and give a \((1+\varepsilon )\)-speed \((1/\varepsilon +3)\)-competitive algorithm.

In this column, we will discuss some papers in online algorithms that appeared in 2021. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

In this column, we will discuss some papers in online algorithms that appeared in 2020. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

One of the most significant algorithmic challenges in the "big data era" is handling instances that are too large to be processed by a single machine. The common practice in this regard is to partition the massive problem instance into smaller ones and process each one of them separately. In some cases, the solutions for the smaller instances are later on assembled into a solution for the whole instance, but in many cases this last stage cannot be pursued (e.g., because it is too costly, because of locality issues, or due to privacy considerations). Motivated by this phenomenon, we consider the following natural combinatorial question: Given a bin-packing instance (namely, a set of items with sizes in (0, 1] that should be packed into unit capacity bins) I and a partition {I-i}(i) of I into clusters, how large is the ratio Sigma(i) Opt(I-i)/Opt(I), where Opt(J) denotes the optimal number of bins into which the items in J can be packed? In this paper, we investigate the supremum of this ratio over all instances I and partitions {I-i}(i), referred to as the bin-packing price of clustering (PoC). It is trivial to observe that if each cluster contains only one tiny item (and hence, Opt(I-i) = 1), then the PoC is unbounded. On the other hand, a relatively straightforward argument shows that under the constraint that Opt(I-i) >= 2, the PoC is 2. Our main challenge was to determine whether the PoC drops below 2 when Opt(I-i) > 2. In addition, one may hope that lim(k ->infinity) PoC(k) = 1, where PoC(k) denotes the PoC under the restriction to clusters I-i with Opt(I-i) >= k. We resolve the former question affirmatively and the latter one negatively: Our main results are that PoC(k) <= 1.951 for any k >= 3 and lim(k ->infinity) PoC(k) = 1.691... Moreover, the former bound cannot be significantly improved as PoC(3) > 1.933. In addition to the immediate contribution of this combinatorial result to "big data" kind of applications, it turns out that it is useful also for an interesting online problem called bin-packing with delays.

In this column, we will discuss some papers in online algorithms that appeared in 2019. As usual, we make no claim at complete coverage here, and have instead made a selection. If we have unaccountably missed your favorite paper and you would like to write about it or about any other topic in online algorithms, please don't hesitate to contact us!

For this issue, Pavel Vesely has contributed a wonderful overview of the ideas that were used in his SODA paper on packet scheduling with Marek Chrobak, Lukasz Jez and Jiri Sgall. This is a problem for which a 2-competitive algorithm as well as a lower bound of ϕ ≈ 1:618 was known already twenty years ago, but which resisted resolution for a long time. It is great that this problem has nally been resolved and that Pavel was willing to explain more of the ideas behind it for this column. He also provides an overview of open problems in this area.